Question

If A = $\left[\begin{array}{ccc}1&-2&3\\-4&2&5\end{array}\right]$ and B = $\left[\begin{array}{ccc}2&3\\4&5\\2&1\end{array}\right]$ , do AB and BA exist? If they exist find them. Do A and B commute with respect to multiplication?

Answer 1

Answer:

$=\left[\begin{array}{cc}0&-4\\10&3\end{array}\right]$

Step-by-step explanation:

$A = \left[\begin{array}{ccc}1&-2&3\\-4&2&5\end{array}\right]$

$B = \left[\begin{array}{ccc}2&3\\4&5\\2&1\end{array}\right]$

AB exist but BA does not exist

A and B do not commute with respect to multiplication

Find AB:

$\left[\begin{array}{ccc}1&-2&3\\-4&2&5\end{array}\right] \left[\begin{array}{ccc}2&3\\4&5\\2&1\end{array}\right]$

$= \left[\begin{array}{ccc}1 \times 2 + (-2) \times 4 + 3 \times 2&1 \times 3 + (-2) \times 5 + 3 \times 1\\(-4) \times 2 + 2 \times 4 + 5 \times 2&(-4) \times 3 + 2 \times 5 + 5 \times 1\end{array}\right]$

$=\left[\begin{array}{cc}0&-4\\10&3\end{array}\right]$